Partial generalizations of some Conjectures in locally symmetric Lorentz spaces

TL;DR

This paper introduces new curvature estimates and partial generalizations of conjectures for spacelike hypersurfaces in locally symmetric Lorentz spaces, expanding understanding of their geometric properties.

Contribution

It defines a new notion for linear Weingarten spacelike hypersurfaces and provides modified curvature operators with estimates, leading to partial conjecture generalizations.

Findings

New estimates for $L(nH)$ and $oxempty(nH)$ on hypersurfaces.

Introduction of a modified Cheng-Yau operator $L$ for curvature analysis.

Partial generalizations of existing conjectures in Lorentz spaces.

Abstract

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying some curvature conditions. By modifying Cheng-Yau's operator $\square$ given in {\cite{ChengYau77}}, we introduce a modified operator $L$ and give new estimates of $L(nH)$ and $\square(nH)$ of such spacelike hypersurfaces. Finally, we give partial generalizations of some conjectures in locally symmetric Lorentz spaces $L_{1}^{n+1}$.